Relative Faber--Krahn inequalities and Trudinger's equation on Riemannian Manifolds

Abstract

We consider on Riemannian manifolds the Trudinger equation \begin{equation*}\partial _{t}u=Δ_{p}u^{\frac{1}{p-1}},\end{equation*} where $p>1$. We prove that a relative $p$--Faber--Krahn inequality is equivalent to the conjunction of volume doubling and a sub-Gaussian upper estimate for non-negative bounded weak subsolutions of Trudinger's equation. We also derive an improved long-time upper estimate under a uniform $p$--Faber--Krahn inequality.